arXiv 2009.054592020-09gr-qchep-thmath-phmath.MP

正则变分完备化与四维Gauss-Bonnet引力

Canonical variational completion and 4D Gauss-Bonnet gravity

Manuel Hohmann, Christian Pfeifer, Nicoleta Voicu

备注：: 16 pages
发表信息：: The European Physical Journal Plus volume 136, Article number: 180 (2021)
DOI：: 10.1140/epjp/s13360-021-01153-0

摘要：

最近，由作用量中的重整化Gauss-Bonnet项得到二阶导数对$D = 4$维引力场方程的有限贡献的建议受到了广泛的关注.它引发了一场讨论，是否雇用重整化程序产生一个定义良好的理论。其中一个主要的批评是基于这样一个事实，即由此产生的场方程不能获得作为欧拉-拉格朗日方程从一个同态不变的行动。在这项工作中，我们使用的技术，从变分的逆演算指出，重整化截断高斯-邦纳方程不能得到任何行动在所有（无论是同态不变或不），在任何维度。然后，我们采用正则变分完成，基于Vainberg-Tonti拉格朗日的概念-其中包括在一个给定的方程组中添加一个正则定义的校正项，以便使它们可从一个动作导出。为了将这种方法应用于建议的$4$ D重整化Gauss-Bonnet方程，我们将变分完备化算法扩展到某些类PDE系统，其中通常的积分提供Vainberg-Tonti拉格朗日发散。我们发现，在$D>4$建议的场方程可以变分完成，选择度量或其逆作为场变量，这两种方法产生一致的相同的拉格朗日，其变化导致四阶场方程。在$D=4$中，变分完备理论的拉格朗日量在两种情况下都是发散的。

英文摘要：

Recently, a proposal to obtain a finite contribution of second derivative order to the gravitational field equations in $D = 4$ dimensions from a renormalized Gauss-Bonnet term in the action has received a wave of attention. It triggered a discussion whether the employed renormalization procedure yields a well-defined theory. One of the main criticisms is based on the fact that the resulting field equations cannot be obtained as the Euler-Lagrange equations from a diffeomorphism invariant action. In this work, we use techniques from the inverse calculus of variations to point out that the renormalized truncated Gauss-Bonnet equations cannot be obtained from any action at all (either diffeomorphism invariant or not), in any dimension. Then, we employ canonical variational completion, based on the notion of Vainberg-Tonti Lagrangian - which consists in adding a canonically defined correction term to a given system of equations, so as to make them derivable from an action. To apply this technique to the suggested $4$D renormalized Gauss-Bonnet equations, we extend the variational completion algorithm to some classes of PDE systems for which the usual integral providing the Vainberg-Tonti Lagrangian diverges. We discover that in $D>4$ the suggested field equations can be variationally completed, choosing either the metric or its inverse as field variables; both approaches yield consistently the same Lagrangian, whose variation leads to fourth order field equations. In $D=4$, the Lagrangian of the variationally completed theory diverges in both cases.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文